On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in R2

Summary In this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior Dirichlet problem in the plane. Specifically, the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region, and the Laplace equation in the corresponding unbounded exterior region, in addition to appropriate boundary and transmission conditions. The main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation. We provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence, uniqueness and approximation results are established. Then, we consider the case where the corresponding operator is strongly monotone and Lipschitz-continuous, and derive asymptotic error estimates for a boundary-finite element solution. We prove the unique solvability of the discrete operator equations, and based on a Strang type abstract error estimate, we show the strong convergence of the approximated solutions. Moreover, under additional regularity assumptions on the solution of the continous operator equation, the asymptotic rate of convergenceO (h) is obtained.The first author's research was partly supported by the U.S. Army Research Office through the Mathematical Science Institute of Cornell University, by the Universidad de Concepción through the Facultad de Ciencias, Dirección de Investigación and Vicerretoria, and by FONDECYT-Chile through Project 91-386.     

An application of the Kantorovich theoremto nonlinear finite element analysis

Summary. Finite element solutions of strongly nonlinear elliptic boundary value problems are considered. In this paper, using the Kantorovich theorem, we show that, if the Fréchet derivative of the nonlinear operator defined by the boundary value problem is an isomorphism at an exact solution, then there exists a locally unique finite element solution near the exact solution. Moreover, several a priori error estimates are obtained. Received March 2, 1998 / Published online September 7, 1999     

Two-grid finite volume element method for linear and nonlinear elliptic problems

Two-grid finite volume element discretization techniques, based on two linear conforming finite element spaces on one coarse and one fine grid, are presented for the two-dimensional second-order non-selfadjoint and indefinite linear elliptic problems and the two-dimensional second-order nonlinear elliptic problems. With the proposed techniques, solving the non-selfadjoint and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem on the fine space and solving the non-selfadjoint and indefinite elliptic problem on a much smaller space; solving a nonlinear elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic problem on a much smaller space. Convergence estimates are derived to justify the efficiency of the proposed two-grid algorithms. A set of numerical examples are presented to confirm the estimates. The work is supported by the National Natural Science Foundation of China (Grant No: 10601045).     

Local a-posteriori error indicators for the Galerkin discretization of boundary integral equations

In this paper we present local a-posteriori error indicators for the Galerkin discretization of boundary integral equations. These error indicators are introduced and investigated by Babuška-Rheinboldt [3] for finite element methods. We transfer them from finite element methods onto boundary element methods and show that they are reliable and efficient for a wide class of integral operators under relatively weak assumptions. These local error indicators are based on the computable residual and can be used for controlling the adaptive mesh refinement. Received March 4, 1996 / Revised version received September 25, 1996     

Finite element approximationof a nonlinear elliptic equation arising from bimaterial problemsin elastic-plastic mechanics

Summary. In [13], a nonlinear elliptic equation arising from elastic-plastic mechanics is studied. A well-posed weak formulation is established for the equation and some regularity results are further obtained for the solution of the boundary problem. In this work, the finite element approximation of this boundary problem is examined in the framework of [13]. Some error bounds for this approximation are initially established in an energy type quasi-norm, which naturally arises in degenerate problems of this type and proves very useful in deriving sharper error bounds for the finite element approximation of such problems. For sufficiently regular solutions optimal error bounds are then obtained for some fully degenerate cases in energy type norms. Received June 12, 1998 / Revised version received June 21, 1999 / Published online June 8, 2000     

The artificial boundary conditions for incompressible materials on an unbounded domain

Summary. In this paper we consider the numerical simulations of the incompressible materials on an unbounded domain in . A series of artificial boundary conditions at a circular artificial boundary for solving incompressible materials on an unbounded domain is given. Then the original problem is reduced to a problem on a bounded domain, which be solved numerically by a mixed finite element method. The numerical example shows that our artificial boundary conditions are very effective. ReceivedJune 7, 1995 / Revised version received August 19, 1996     

Finite element approximation of a semilinear elliptic problem with a singular nonlinearity

Given , we consider the following problem: find , such that where or 3, and in . We prove and error bounds for the standard continuous piecewise linear Galerkin finite element approximation with a (weakly) acute triangulation. Our bounds are nearly optimal. In addition, for d = 1 and 2 and we analyze a more practical scheme involving numerical integration on the nonlinear term. We obtain nearly optimal and error bounds for d = 1. For this case we also present some numerical results. Received July 4, 1996 / Revised version received December 18, 1997     

Error estimates for the finite element solution of elliptic problems with nonlinear newton boundary conditions

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition. The existence and uniqueness of the solution of the continuous pioblem is a consequence of the monotone operator theory. The main attention is paid to the investigation of the finite element approximation using numeriral integration for the evaluation of boundary integrals. The error estimates for the solution of the discrete finite element problem are derived     

Composite finite elements for the approximation of PDEs on domains with complicated micro-structures

Summary. Usually, the minimal dimension of a finite element space is closely related to the geometry of the physical object of interest. This means that sometimes the resolution of small micro-structures in the domain requires an inadequately fine finite element grid from the viewpoint of the desired accuracy. This fact limits also the application of multi-grid methods to practical situations because the condition that the coarsest grid should resolve the physical object often leads to a huge number of unknowns on the coarsest level. We present here a strategy for coarsening finite element spaces independently of the shape of the object. This technique can be used to resolve complicated domains with only few degrees of freedom and to apply multi-grid methods efficiently to PDEs on domains with complex boundary. In this paper we will prove the approximation property of these generalized FE spaces. Received June 9, 1995 / Revised version received February 5, 1996     

On the adaptive coupling of FEM and BEM in 2–d–elasticity

Summary. This paper concerns the combination of the finite element method (FEM) and the boundary element method (BEM) using the symmetric coupling. As a model problem in two dimensions we consider the Hencky material (a certain nonlinear elastic material) in a bounded domain with Navier–Lamé differential equation in the unbounded complementary domain. Using some boundary integral operators the problem is rewritten such that the Galerkin procedure leads to a FEM/BEM coupling and quasi–optimally convergent discrete solutions. Beside this a priori information we derive an a posteriori error estimate which allows (up to a constant factor) the error control in the energy norm. Since information about the singularities of the solution is not available a priori in many situation and having in mind the goal of an automatic mesh–refinement we state adaptive algorithms for the –version of the FEM/BEM–coupling. Illustrating numerical results are included. Received April 15, 1994 / Revised version received January 8, 1996     

On the finite element approximation of a cascade flow problem

Summary The finite element analysis of a cascade flow problem with a given velocity circulation round profiles is presented. The nonlinear problem for the stream function with nonstandard boundary conditions is discretized by conforming linear triangular elements. We deal with the properties of the discrete problem and study the convergence of the method both for polygonal and nonpolygonal domains, including the effect of numerical integration.     

Multilevel methods for mixed finite elements in three dimensions

In this paper we consider second order scalar elliptic boundary value problems posed over three–dimensional domains and their discretization by means of mixed Raviart–Thomas finite elements [18]. This leads to saddle point problems featuring a discrete flux vector field as additional unknown. Following Ewing and Wang [26], the proposed solution procedure is based on splitting the flux into divergence free components and a remainder. It leads to a variational problem involving solenoidal Raviart–Thomas vector fields. A fast iterative solution method for this problem is presented. It exploits the representation of divergence free vector fields as s of the –conforming finite element functions introduced by Nédélec [43]. We show that a nodal multilevel splitting of these finite element spaces gives rise to an optimal preconditioner for the solenoidal variational problem: Duality techniques in quotient spaces and modern algebraic multigrid theory [50, 10, 31] are the main tools for the proof. Received November 4, 1996 / Revised version received February 2, 1998     

Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy

Summary. A model for the phase separation of a multi-component alloy with non-smooth free energy is considered. An error bound is proved for a fully practical piecewise linear finite element approximation using a backward Euler time discretization. An iterative scheme for solving the resulting nonlinear algebraic system is analysed. Finally numerical experiments with three components in one and two space dimensions are presented. In the one dimensional case we compare some steady states obtained numerically with the corresponding stationary solutions of the continuous problem, which we construct explicitly. Received September 28, 1995 / Revised version received May 6, 1996     

Finite element simulations of window Josephson junctions

This paper deals with the numerical simulation of the steady state two dimensional window Josephson junctions by finite element method. The model is represented by a sine-Gordon type composite PDE problem. Convergence and error analysis of the finite element approximation for this semilinear problem are presented. An efficient and reliable Newton-preconditioned conjugate gradient algorithm is proposed to solve the resulting nonlinear discrete system. Regular solution branches are computed using a simple continuation scheme. Numerical results associated with interesting physical phenomena are reported. Interface relaxation methods, which by taking advantage of special properties of the composite PDE, can further reduce the overall computational cost are proposed. The implementation and the associated numerical experiments of a particular interface relaxation scheme are also presented and discussed.     

Consistency,stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems

Summary. In an abstract framework we present a formalism which specifies the notions of consistency and stability of Petrov-Galerkin methods used to approximate nonlinear problems which are, in many practical situations, strongly nonlinear elliptic problems. This formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems. This theory is illustrated with the example: in a two dimensional domain with Dirichlet boundary conditions. Received June 10, 1992 / Revised version received February 28, 1994     

Integration of singular Galerkin-type boundary element integrals for 3D elasticity problems

Summary. A Galerkin approximation of both strongly and hypersingular boundary integral equation (BIE) is considered for the solution of a mixed boundary value problem in 3D elasticity leading to a symmetric system of linear equations. The evaluation of Cauchy principal values (v. p.) and finite parts (p. f.) of double integrals is one of the most difficult parts within the implementation of such boundary element methods (BEMs). A new integration method, which is strictly derived for the cases of coincident elements as well as edge-adjacent and vertex-adjacent elements, leads to explicitly given regular integrand functions which can be integrated by the standard Gauss-Legendre and Gauss-Jacobi quadrature rules. Problems of a wide range of integral kernels on curved surfaces can be treated by this integration method. We give estimates of the quadrature errors of the singular four-dimensional integrals. Received June 25, 1995 / Revised version received January 29, 1996     

Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls

Summary. An optimal control problem for impressed cathodic systems in electrochemistry is studied. The control in this problem is the current density on the anode. A matching objective functional is considered. We first demonstrate the existence and uniqueness of solutions for the governing partial differential equation with a nonlinear boundary condition. We then prove the existence of an optimal solution. Next, we derive a necessary condition of optimality and establish an optimality system of equations. Finally, we define a finite element algorithm and derive optimal error estimates. Received March 10, 1993 / Revised version received July 4, 1994     

Fast parallel solvers for symmetric boundary element domain decomposition equations

Summary. The boundary element method (BEM) is of advantage in many applications including far-field computations in magnetostatics and solid mechanics as well as accurate computations of singularities. Since the numerical approximation is essentially reduced to the boundary of the domain under consideration, the mesh generation and handling is simpler than, for example, in a finite element discretization of the domain. In this paper, we discuss fast solution techniques for the linear systems of equations obtained by the BEM (BE-equations) utilizing the non-overlapping domain decomposition (DD). We study parallel algorithms for solving large scale Galerkin BE–equations approximating linear potential problems in plane, bounded domains with piecewise homogeneous material properties. We give an elementary spectral equivalence analysis of the BEM Schur complement that provides the tool for constructing and analysing appropriate preconditioners. Finally, we present numerical results obtained on a massively parallel machine using up to 128 processors, and we sketch further applications to elasticity problems and to the coupling of the finite element method (FEM) with the boundary element method. As shown theoretically and confirmed by the numerical experiments, the methods are of algebraic complexity and of high parallel efficiency, where denotes the usual discretization parameter. Received August 28, 1996 / Revised version received March 10, 1997     

Dual-mixed finite element approximation of Stokes and nonlinear Stokes problems using trace-free velocity gradients

In this work a finite element method for a dual-mixed approximation of Stokes and nonlinear Stokes problems is studied. The dual-mixed structure, which yields a twofold saddle point problem, arises in a formulation of this problem through the introduction of unknown variables with relevant physical meaning. The method approximates the velocity, its gradient, and the total stress tensor, but avoids the explicit computation of the pressure, which can be recovered through a simple postprocessing technique. This method improves an existing approach for these problems and uses Raviart-Thomas elements and discontinuous piecewise polynomials for approximating the unknowns. Existence, uniqueness, and error results for the method are given, and numerical experiments that exhibit the reduced computational cost of this approach are presented.     

An optimal order convergence for a weak formulation of the compressible Stokes system with inflow boundary condition

Summary. We formulate the compressible Stokes system given in (1.1) into a (new) weak formulation (2.1). A finite element method for this is presented. Existence and uniqueness of the finite element method is shown. An optimal error estimate for the numerical approximation is obtained. Numerical examples are given, showing its efficiency and rates of convergence of the approximate solutions that results from the discrete problem (3.1). Received October 20, 1996 / Revised version received January 21, 1999 / Published online: April 20, 2000